Abstract:
The digitalized world needs efficient and reliable algorithms to solve problems in economy and society. Trust and security can only be guaranteed by constructive proofs. Therefore, applied mathematics and computer science should be combined with foundational research of logic and proof theory. Foundational research in proof mining, program extraction, and reverse mathematics is following this line. These foundational programs stand in elder philosophical traditions of constructive and intuitionistic mathematics. Current foundational programs (e.g., HoTT) are inspired by the idea of a proof-checking software to guarantee trust and security in mathematics. These are not only theoretical demands. They are deeply connected with great challenges of our digitalized world: Current AI-technology (e.g., machine learning, autonomous car driving, and financial mathematics) also needs more explainability and provability, in order to improve trust, reliability and security in society.